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volume 7, issue 3, article 100, 2006.

Received 07 April, 2005;

accepted 07 April, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

A NOTE ON AN INDIVIDUAL BIOEQUIVALENCE SETTING

C. SURULESCU AND N. SURULESCU

Institut für Angewandte Mathematik INF 294, 69120 Heidelberg Germany.

EMail:Christina.Surulescu@iwr.uni-heidelberg.de EMail:Nicolae.Surulescu@iwr.uni-heidelberg.de

2000c Victoria University ISSN (electronic): 1443-5756 079-06

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A Note on an Individual Bioequivalence Setting C. Surulescu and N. Surulescu

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J. Ineq. Pure and Appl. Math. 7(3) Art. 100, 2006

http://jipam.vu.edu.au

Abstract

We give a new simpler proof along with a generalization for the inequality of Yao and Iyer [10] arising in bioequivalence studies and by using a nonparametric approach we also discuss an extension of the individual bioequivalence setting to the case where the data are not necessarily normally distributed.

2000 Mathematics Subject Classification:60E15, 62F03, 62P10.

Key words: Bioequivalence, Probabilistic inequalities

1. Introduction

Bioequivalence testing is required when trying to get the approval for manufacturing and selling of a generic drug having mainly the same properties with a (more expensive) reference (brand-name) drug. Establishing bioequivalence saves the generic drug manufacturer from performing expensive clinical trials to demonstrate the quality of his product. Two drugs are considered bioequivalent if they are absorbed into the blood and become active at about the same rate and concentration. Bioequivalent drugs are supposed to provide the same therapeutic effect.

For explaining the notions and notations we use, we recall the problem setting in [10] for the individual bioequivalence. Thus, the amount of the chemical absorbed by a patient’s bloodstream when using a reference drug is a random variable X, which has mean µ_R and standard deviation σ_R. The corresponding variable for the same patient when using the generic drug has meanµT and standard deviationσ_T. The therapeutic window of a patient is defined to be the interval in which must lie the concentration of the chemical in the bloodstream, in order for the drug to be classified as beneficial for that patient. Usually, the therapeutic window is assumed to be an interval centered at the mean µ_R, namely the range (µ_R −zσ_R, µ_R +zσ_R). The drug will be uneffective if the amount absorbed in the bloodstream is too low and it could cause severe side effects if too much of the chemical substance is absorbed.

Denoting by p_R and p_T the probabilities that the subject will have benefit from using drug R, respectively T, the regulatory agency might approve the marketing of drugT provided that ^p_p^T

R ≥γ, whereγ is about 1 or even larger.

The therapeutic window of a patient is generally unknown, therefore a quan-

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tity of interest for the approval procedure will beinfz≥0 pT

pR.

Usually, to derive this quantity the assumption that X and T have normal distributions is made, though it is well known that in practice this is rarely the case. Under this assumption, Yao and Iyer [10] have shown that

z>0inf p_T p_R = inf

z>0

Φ(^µ^R^+zσ_σ^R^−µ^T

T )−Φ(^µ^R^−zσ_σ^R^−µ^T

T )

Φ(z)−Φ(−z) (1.1)

= min (

1,σ_R√ 2π σ_T φ

µ_T −µ_R σ_T

) ,

whereφ andΦare respectively the probability density function and the cumu- lative density function of a standard normal variable.

Thus, the approval of manufacturing the generic drug T may be granted if from the statistical analysis of experimental data it can be proven that

(1.2) σR

√2π σ_T φ

µT −µR

σ_T

≥γ.

Alternatively to (1.2), a more flexible approval criterion can be used, namely one of the type

(1.3) `(Z_γ)≥b_γ,

for some large enough given bound bγ, whereZγ := {z > 0 : _p^p^T

R > γ} and

` is for instance the Lebesgue measure. We will discuss in the next sections sufficient conditions for (1.3) to be satisfied.

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For more information about individual bioequivalence see [1], [7], [8], [4], [5], [6] and the references therein.

In this paper we give a simpler proof and a generalization of the inequality of Yao and Iyer and we also give a nonparametric extension of the above bioequivalence setting.

For the sake of clarity, we put all our proofs in the Appendix.

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2. A Generalization of the Inequality of Yao and Iyer

The main result of Yao and Iyer in [10] was the proof of the following inequality:

(2.1) Φ(^z−µ_σ )−Φ(^−z−µ_σ )

Φ(z)−Φ(−z) >min (

1,

√2π σ φµ

σ

) ,

for allz >0,µ∈R\{0}andσ∈(0,∞)\{1}; this inequality comes from (1.1) after some changes of notations.

As in [10], observe that it is enough to treat the caseµ ≥ 0. Here we will prove the following generalization:

Proposition 2.1. In the above settings, we have:

(i) Ifσ >1, then

Φ(^z−µ_σ )−Φ(^−z−µ_σ )

Φ(z)−Φ(−z) > e⁻^µ

2

2σ2Φ(^z_σ)−Φ(^−z_σ ) Φ(z)−Φ(−z)

> e⁻^µ

2 2σ2 1

σ

1 + 1 6

1− 1

σ²

·z²e⁻^z

2 2

> 1 σe⁻^µ

2

2σ2 = min (

1,

√2π σ φµ

σ

)

, ∀z >0.

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(ii) Ifσ ∈(0,1)then Φ(^z−µ_σ )−Φ(^−z−µ_σ )

Φ(z)−Φ(−z) ≥min (

1,

√2π σ φµ

σ

) Φ(^z−µ(σ)_σ )−Φ(^{−z−µ(σ)}_σ ) Φ(z)−Φ(−z)

>min (

1,

√2π σ φµ

σ

)

, ∀z >0, ∀µ > 0, (2.2)

whereµ(σ) := σ q

2 ln_σ¹.

This result is based on Lemma2.2, generalising Lemma 3 in [10], (which was the most difficult part of the proof therein) and on Lemmas 2.3 and 2.4 below.

Lemma 2.2. The function σ 7→ F(σ) := Φ(^z−µ(σ)_σ )−Φ(^{−z−µ(σ)}_σ ) is strictly decreasing on(0,1), withµ(σ)as above.

Lemma 2.3. For everyσ >0,σ6= 1, we have:

(2.3) Φ _σ^z

−Φ ^−z_σ

Φ(z)−Φ(−z) −min

1, 1 σ

> 1

6σ 1−min

1, 1 σ

2!

·z²e⁻^z

2 2

+ 1

σ −min

1,1 σ

e⁻^z

2

2σ2 >0, ∀z >0.

The following lemma generalises Lemma 1 and Lemma 4 in [10]:

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Lemma 2.4.

(2.4) Φ(z−µ₁)−Φ(−z−µ₁)

Φ(z−µ₂)−Φ(−z−µ₂) >min n

1, e⁻¹²^[(µ¹⁾²^−(µ²⁾²^] o

∀z >0,µ₁, µ₂ ∈R,µ₁ 6=µ₂.

Remark 1. Replacingzwith _σ^z,µ₁ by ^µ_σ¹ andµ₂ by ^µ_σ² then

• forµ₁ > µ₂ >0one has Φ ^z−µ_σ ¹

−Φ ^−z−µ_σ ¹ Φ ^z−µ_σ ²

−Φ ^−z−µ_σ ² >min

1, e⁻¹²

h(^µσ¹)²⁻(^µσ²)²ⁱ

=e⁻¹²

h(^µσ¹)²⁻(^µσ²)²ⁱ, thus the function

(0,∞)3µ7→B₁(µ) :=e¹²⁽^µ^σ⁾²

Φ

z−µ σ

−Φ

−z−µ σ

is increasing, i.e. Lemma 1 in [10] is obtained;

• forµ2 > µ1 >0one has

Φ(^z−µ_σ ¹)−Φ(^−z−µ_σ ¹) Φ(^z−µ_σ ²)−Φ(^−z−µ_σ ²) >1, thus the function

(0,∞)3µ7→B₂(µ) := Φ

z−µ σ

−Φ

−z−µ σ

is proved to be decreasing, i.e. we obtained Lemma 4 in [10].

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Remark 2. Sufficient conditions for (1.3) to be satisfied can be easily derived upon using forσ > 1the monotonicity of the functionz 7→z²e⁻^z

2

2 (increasing on(0,√

2), decreasing on[√

2,∞)) and for0< σ <1the fact that Φ ^z−µ_σ

−Φ ^−z−µ_σ

Φ(z)−Φ(−z) >min (

1,

√2π σ φµ

σ

)Φ_z−µ(σ)

σ

−Φ_{−z−µ(σ)}

σ

Φ(z)−Φ(−z)

≥σmin (

1,

√2π σ φµ

σ

)Φ(_σ^z)−Φ(^−z_σ )

Φ(z)−Φ(−z), ∀z >0 and this term can be minorated by applying Lemma2.3.

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3. A Nonparametric Approach for the Bioequivalence Setting

Consider now that the random variables X, T from the Introduction have continuous univariate distributions, with the densities f_X, respectively f_T, which are not necessarily Gaussian, and assume that we correspondingly have the in- dependent observations: x₁, . . . , x_m, respectively t₁, . . . , t_n. Then f_X and f_T can be estimated by using the classical nonparametric estimators:

fˆ_X(x) = 1 mh^X_m ·

m

X

i=1

K

x−x_i h^X_m

and

fˆ_T(t) = 1 nh^T_n ·

n

X

i=1

K

t−t_i h^T_n

,

whereK is a kernel,h^X_m andh^T_n are the bandwidths (with the usual properties:

h^X_m → 0, h^T_n → 0form, n→ ∞;mh^X_m → 0, nh^T_n → ∞form, n→ ∞and, of course,h^X_mandh^T_n have to be chosen in practice by a corresponding criterion, see [9]).

With the above notations, the fractions of interest for bioequivalence studies are

p_T

p_R =R(z) :=

µR+zσR

R

µR−zσ_R

fT(t)dt

µR+zσR

R

µR−zσ_R

f_X(x)dx (3.1)

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≈R(z) :=ˆ

µR+zσR

R

µR−zσ_R

fˆ_T(t)dt

µR+zσR

R

µR−zσR

fˆ_X(x)dx

, z >0,

and for the approval procedure it will be important to find the quantity of interest inf_z>0R(z), but this is clearly more difficult than for the case presentedˆ in the previous section and an analytic treatment is hardly possible. Even when considering the Gaussian kernel, the available nonlinear optimization proce- dures are surprisingly very time consuming. However, a great (and also easy to implement) simplification can be achieved when using one of the following kernels (see e.g., [9]):

• the rectangular kernel:K(u) := ¹₂χ(−1,1)(u);

• the triangular kernel: K(u) := (1− |u|)χ(−1,1)(u);

• the Epanechnikov kernel:K(u) := ³₄(1−u²)χ(−1,1)(u);

• the triangle kernel: K(u) := (1− |u|)χ(−1,1)(u);

• the double Epanechnikov kernel:K(u) := 3|u|(1− |u|)χ_(−1,1)(u).

Moreover, the procedure can also be used for finding the largest therapeutic window under which the fraction of interest exceeds some given positive constant γ. Since all kernels above are supported on (−1,1), then the global minimum can be found in the following way:

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• constructA^T,+_n :=A^T_n ∩(0,∞), where

A^T_n :=

n

[

i=1

−ti−h^T_n −µR

σ_R , ti +h^T_n −µR

σ_R ,

t_i−h^T_n −µ_R

σ_R , −t_i+h^T_n−µ_R σ_R

;

• constructA^X,+_m :=A^X_m∩(0,∞), where

A^X_m :=

m

[

j=1

−x_j−h^X_m−µ_R

σ_R , x_j +h^X_m−µ_R

σ_R ,

x_j−h^X_m−µ_R

σ_R , −x_j +h^X_m−µ_R σ_R

;

• ifK is not the rectangular kernel, then construct A⁺_T,deriv as the set con- stituted by the critical points ofR(z)ˆ on the union of the subintervals of R+ whereRˆ is differentiable (which in this case are roots of some polynomials, thus easy to be handled by the computer); ifK is the rectangular kernel, setA⁺_T,deriv =∅.

• denoteA:=A^T,+_n S

A^X,+_m S

A⁺_T,deriv. Then we have:

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Proposition 3.1. With the notations above and withK being one of the kernels enumerated before,

R(z) =ˆ

1 nh^T_n

n

P

i=1

µR+zσR

R

µR−zσ_R

K

t−ti

h^T_n

dt

1 mh^X_m

m

P

i=1

µR+zσR

R

µR−zσ_R

K x−xi

h^X_m

dx

≥minn

R(zˆ _max),lim

z→0

R(z),ˆ R(A)ˆ o ,

wherez_max = max{|ζ| : ζ ∈ A},

limz→0

R(z) =ˆ mh^X_m nh^T_n

n

P

i=1

K

µR−t_i h^T_n

m

P

i=1

K

µR−x_i h^X_m

for the continuous kernels above,

and

z→0lim

R(z) =ˆ mh^X_m nh^T_n

n

P

i=1

h K₊

µR−ti

h^T_n

+K−

µR−ti

h^T_n

i

m

P

i=1

h K₊

µR−x_i h^X_m

+K₋

µR−x_i h^X_m

i for the rectangular kernel,

with the notationsK₊(ξ) := lim_t&ξK(t), respectivelyK₋(ξ) := lim_t%ξK(t).

Remark 3. IfK is one of the kernels presented above, then for eachγ >0and using an algorithm similar to the one described above one can easily determine Z_γ, in order to check the approval condition (1.3).

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Simulation result: Figure 1illustrates a simulation with m = 150, n = 160 for the case where X and T are normally distributed and the nonparametric estimators are constructed with the Epanechnikov kernel. The continuous lines representR(z)ˆ and its minimum, which is 0.012; the lines of circles represent R(z)and its minimum, which is 0.0088. This shows that the Gaussian case can be well recovered with this nonparametric procedure.

1

0.8

0.6

0.4

0.2

0

z

20 16 12 8

4

Figure 1: The continuous line isR(z); the line of circles isˆ R(z); the horizontal lines are the corresponding minimums

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4. Conclusions

In this paper we gave a generalization and a simpler proof of the inequality of Yao and Iyer [10] concerning an individual bioequivalence setting when the data were supposed to be normally distributed. Our generalization can be used to develop a more flexible approval criterion (of the type (1.3)) for manufacturing and selling a drug which is supposed to be bioequivalent with a reference one. Finally, we extended these settings to the more general situation when the data are not necessarily normally distributed, upon using the nonparametric estimation technique. This idea can also be useful in the context of global nonlinear optimization problems.

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A. Proofs

Proof of Lemma2.2. Denotingθ := _σ¹ ∈(1,∞), we have Φ

z−µ(σ) σ

−Φ

−z−µ(σ) σ

= Φ(zθ−√

2 lnθ)−Φ(−zθ−√ 2 lnθ).

Considerg : (1,∞)→R, g(θ) := Φ(zθ−√

2 lnθ)−Φ(−zθ−√

2 lnθ), ∀θ >

1. Observe that g⁰(θ) =

z+ 1

θ√ 2 lnθ

·φ(zθ−√ 2 lnθ)

"

zθ√

2 lnθ−1 zθ√

2 lnθ+ 1 + exp(−2zθ√ 2 lnθ)

#

>0,

for allθ >1, becauseu:=zθ√

2 lnθ >0and it is easy to see that u−1

u+ 1 +e^−2u >0, ∀u >0.

Proof of Lemma2.3. Ifσ > 1, we can write Φz

σ −Φ

−z σ

= 1

σ√ 2π

Z z

−z

e⁻^x

2 2σ2dx

> 1 σ

Φ(z)−Φ(−z) + 1

√2π · 1 2

1− 1

σ² Z z

−z

x²e⁻^x

2 2 dx

,

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sincee^x

2

2 (¹⁻_σ¹2) ≥1 +^x₂² 1− _σ¹2

,∀x∈R\{0}. Further, from the decreasing monotonicity ofx7→e⁻^x

2

2 on[0, z]deduce that Φz

σ −Φ

−z σ

≥ 1 σ

Φ(z)−Φ(−z) + 1

√2π

1− 1 σ²

·z³ 3e⁻^z

2 2

and, since

Φ(z)−Φ(−z)< zp

2/π, ∀z >0 (2.3) is proved.

For the case whereσ ∈(0,1)we have:

Φ(^z_σ)−Φ(^−z_σ )

Φ(z)−Φ(−z) = 1 + 2 Φ(_σ^z)−Φ(z) Φ(z)−Φ(−z)

= 1 + 2 R ^z_σ

z

√1 2πe⁻^x

2 2 dx Φ(z)−Φ(−z)

>1 + 2

√1 2πe⁻^z

2

2σ2 ·z _σ¹ −1 Φ(z)−Φ(−z)

>1 + 1

σ −1

e⁻^z

2

2σ2,∀z >0, upon using the same method as in the previous case.

Proof of Lemma2.4. Let

Q(z) := Φ(z−µ₁)−Φ(−z−µ₁)

−e⁻¹²^[(µ¹⁾²^−(µ²⁾²^][Φ(z−µ₂)−Φ(−z−µ₂)], z >0.

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Then

Q⁰(z) = 2

√2πe⁻¹²^[(µ¹⁾²^+z²^]

e^zµ¹ +e^−zµ¹

2 − e^zµ² +e^−zµ² 2

= 2

√2πe⁻¹²^[(µ¹⁾²^+z²^][cosh(z|µ₁|)−cosh(z|µ₂|)], ∀z >0.

Now using the fact that the hyperbolic cosine is an increasing function on (0,∞), we have thatQ⁰(z)>0if|µ1|>|µ2|andQ⁰(z)<0if|µ1|<|µ2|.

Thus, if|µ₁|>|µ₂|, thenQ(z)>0,∀z >0, i.e. inequality (2.4) is satisfied, since the minimum therein ise⁻¹²^[(µ¹⁾²^−(µ²⁾²^].

If|µ₁|<|µ₂|, then consider the function

G(µ) := Φ(z−µ)−Φ(−z−µ)

and observe that it is decreasing on(0,∞), since its derivative is G⁰(µ) =− 1

√2πe^−(z²^+µ²^)/2[e^zµ−e^−zµ]<0, ∀z >0, µ >0.

Observing thatG(µ) = G(−µ),∀µ∈ R, we have thatG(µ₂) = G(|µ₂|) <

G(|µ₁|) =G(µ₁), which proves the inequality in this case.

Proof of Proposition2.1. In the case (i) observe that we can write Φ(^z−µ_σ )−Φ(^−z−µ_σ )

Φ(z)−Φ(−z) = Φ(^z−µ_σ )−Φ(^−z−µ_σ )

Φ(^z_σ)−Φ(^−z_σ ) · Φ(_σ^z)−Φ(^−z_σ ) Φ(z)−Φ(−z).

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Now apply Lemma 2.4for the first term in the right hand side and Lemma2.3 for the other one.

In the case (ii), ifµ ∈(0, µ(σ)), then we have thatB₂(µ) ≥ B₂(µ(σ))(see Lemma2.4and Remark1). Further, use Lemma2.2to obtain the last inequality in (2.2).

Ifµ > µ(σ),thenσB₁(µ)≥σB₁(µ(σ))(see again Lemma2.4and Remark 1). Then Lemma2.2implies that

σB1(µ(σ))

Φ(z)−Φ(−z) = F(σ)

Φ(z)−Φ(−z) >1, which completes the proof.

Proof of Proposition3.1. The proof follows observing that for each of the ker- nels above R(z)ˆ becomes a rapport of polynomials on some finite intervals dictated by the points in A^X,+_m S

A^T,+_n . Thus, the problem reduces to charac- terising the minimum of these fractional expressions on such corresponding finite intervals, which in this particular case means to find the roots of the polynomials at the numerator of the derivatives. It only remains to observe that limz→∞R(z) = ˆˆ R(z_max).

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References

[1] S. ANDERSON AND W.W. HAUCK, Consideration of individual bioe- quivalence, J. of Parmacokinetics and Biopharmaceutics, 18 (1990), 259–

274.

[2] I.G. GRAMA, On moderate deviations for martingales, The Annals of Probability, 25(1) (1997), 152–183.

[3] B. JONES AND M.G. KENWARD, Design and Analysis of Cross-Over Trials (2nd edition), Chapman & Hall, 2003.

[4] B.F.J. MANLY AND R.I.C.C. FRANCIS, Testing for mean and variance differences with samples from distributions that may be non-normal with unequal variances, Journal of Statistical Computation and Simulation, 72(8) (2002), 633–646.

[5] B.F.J. MANLY, One-sided tests of bioequivalence with nonnormal distri- butions and unequal variances, Journal of Agricultural, Biological and En- vironmental Statistics, 9(3) (2004), 270–283.

[6] L.L.MCDONALD, S. HOWLIN, J. POLYAKOVA AND C.J. BIL- BROUGH, Evaluation and Comparison of Hypothesis Testing Techniques for Bond Release Applications, a project for the Abandoned Coal Mine Lands Research Program at the University of Wyoming, 2003.

[7] R. SCHALL, Assessment of individual and population bioequivalence us- ing the probability that bioavailabilities are similar, Biometrics, 51 (1995), 615–626.

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[8] R. SCHALL AND H.G. LUUS, On population and individual bioequiva- lence, Statistics in Medicine, 12 (1993), 1109–1124.

[9] D.W. SCOTT, Multivariate Density Estimation. Theory, Practice, and Vi- sualization, John Wiley & Sons, 1992.

[10] YI-CHING YAO ANDH. IYER, On an inequality for the normal distribu- tion arising in bioequivalence studies, J. Appl. Prob., 36 (1999), 279–286.